To describe a system consisting of a large number of particles in a physically useful manner, recourse must be had to so-called statistical procedures. If the mechanical laws operating in the system are those of classical mechanics, and if the system is sufficiently dilute, the resulting statistical treatment is referred to as Boltzmann or classical statistics. (Dilute in this instance means that the total volume available is much larger than the proper volume of the particles.) A gas is a typical example: The molecules interacting according to the laws of classical mechanics are the constituents of the system, and the pressure, temperature, and other parameters are the overall entities which determine the macroscopic behavior of the gas. In a case of this kind it is neither possible nor desirable to solve the complicated equations of motion of the molecules; one is not interested in the position and velocity of every molecule at any time. The purpose of the statistical description is to extract from the mechanical description just those features relevant for the determination of the macroscopic properties and to omit others.
The basic notion in the statistical description is that of a distribution function. Suppose a system of N molecules is contained in a volume V. The molecules are moving around, colliding with the walls and with each other. Construct the following geometrical representation of the mechanical system. Introduce a six-dimensional space (usually called the μ space), three of its coordinate axes being the spatial coordinates of the vessel x, y, z, and the other three indicating cartesian velocity components υx, υy, υz. A molecule at a given time, having a specified position and velocity, may be represented by a point in this six-dimensional space. The state of the gas, a system of N molecules, may be represented by a cloud of N points in this space. In the course of time, this cloud of N points moves through the μ space.
Note that the μ space is actually finite; the coordinates x, y, z of the molecules' position are bounded by the finite size of the container, and the velocities are bounded by the total energy of the system. Imagine now that the space is divided into a large number of small cells, of sizes w1, … , wi, … . A certain specification of the state of the gas is obtained if, at a given time t, the numbers n1(t), … , ni(t), … of molecules in the cells 1, … , i, … are given. The change in the state of the system in the course of time is now determined by the behavior of the functions ni(t) as functions of time. Strictly speaking, these functions may change discontinuously in the course of time. Just how detailed the description so obtained is depends, of course, on the sizes chosen for the cells wi. One gets the crudest possible description by having just one cell of size V in the coordinate subspace, with N particles in it all the time. A very detailed description is obtained if cells the size of a molecule are chosen. In that case even a small change in the state of the molecules will cause a profound alteration in the numbers ni. To apply statistical methods, one must choose the cells such that on the one hand a cell size w is small compared to the macroscopic dimensions of the system, while on the other hand w must be large enough to allow a large number of molecules in one cell. That this is possible stems from the fact that the molecular dimensions (linear dimension about 10−8 cm) are small compared to macroscopic dimensions (typical linear dimension 102 cm). In this case it is possible to have about 1015 cells, each one of linear dimension 10−3, where in each cell there is “room” for 1015 molecules. If the cells are thus chosen, the numbers ni(t), the occupation numbers, will be slowly changing functions of time. The distribution functions fi(t) are defined by Eq. (1).
The distribution function fi describes the state of the gas, and fi of course varies from cell to cell. Since a cell i is characterized by a given velocity range and position range, and since for appropriately chosen cells f should vary smoothly from cell to cell, f is often considered as a continuous function of the variables x, y, z, υx, υz. The cell size w then can be written as dxdydzdυx dυy dυz.
In applications the continuous notation is often used; this should not obscure the fact that the cells are finite. L. Boltzmann called them physically infinitesimal.
Since a cell i determines both a position and a velocity range, one may associate an energy εi with a cell. This is the energy a single molecule possesses when it has a representative point in cell i. This assumes that, apart from instantaneous collisions, molecules exert no forces on each other. If this were not the case, the energy of a molecule would be determined by the positions of all other molecules.
Boltzmann equation; H theorem
Most of the physically interesting quantities follow from a knowledge of the distribution function; the main problem in Boltzmann statistics is to find out what this function is. It is clear that ni(t) changes in the course of time for three reasons: (1) Molecules located at the position of cell i change their positions and hence move out of cell i; (2) molecules under the influence of outside forces change their velocities and again leave the cell i; and (3) collisions between the molecules will generally cause a (discontinuous) change of the occupation numbers of the cells. Whereas the effect of (1) and (2) on the distribution function follows directly from the mechanics of the system, a separate assumption is needed to obtain the effect of collisions on the distribution function. This assumption, the collision-number assumption, asserts that the number of collisions per unit time, of type (i,j) → (k,l) [molecules from cells i and j collide to produce molecules of different velocities which belong to cells k and l], called Aklij, is given by Eq. (2).
Here aklij depends on the collision configuration and on the size and kind of the molecules but not on the occupation numbers. Furthermore, for a collision to be possible, the conservation laws of energy and momentum must be satisfied; so if ε′i = ½m v 2i and pi = m vi, then Eqs. (3a) and (3b)
hold. It is possible to show that the geometrical factor aklij has the property given by Eq. (4).
Here aklij is the geometrical factor belonging to the collision which, starting from the final velocities (k and l), reproduces the initial ones (i and j). Gains and losses of the molecules in, say, cell i can now be observed. If the three factors causing gains and losses are combined, the Boltzmann equation, written as Eq. (5),
is obtained. Here Δx fi is the gradient of f with respect to the positions, Δυ fi refers similarly to the velocities, and Xi is the outside force per unit mass at cell i. This nonlinear equation determines the temporal evolution of the distribution function. Exact solutions are difficult to obtain. Yet Eq. (5) forms the basis for the kinetic discussion of most transport processes. There is one remarkable general consequence, which follows from Eq. (5). If one defines H(t) as in Eq. (6),
hold. Hence H is a function which in the course of time always decreases. This result is known as the H theorem. The special distribution which is characterized by Eq. (8)
has the property that collisions do not change the distribution in the course of time; it is an equilibrium or stationary distribution
It should be stressed that the form of the equilibrium distribution may be determined from Eq. (8), with the help of the relations given as Eqs. (3a) and (3b). For a gas which as a whole is at rest, it may be shown that the only solution to functional Eq. (8) is given by Eqs. (9a) or (9b).
Here A and β are parameters, not determined by Eq. (8), and U is the potential energy at the point x, y, z. Equations (9a) and (9b) are the Maxwell-Boltzmann distribution. Actually A and β can be determined from the fact that the number of particles and the energy of the system are specified, as in Eqs. (10a) and (10b).
hold. Therefore β is related directed to the energy per particle while A is related to β and to the number density. Other physical quantities, such as pressure, must now be calculated in terms of A and β. Comparing such calculated entities with those observed, at a certain point one identifies an entity like β (a parameter in the distribution function) with a measured quantity, such as temperature. More precisely β = 1/kT, where k is the Boltzmann constant. This is the result of an identification. It is not a deduction. It is possible by specialization of Eq. (9b) to obtain various familiar forms of the distribution law. If U = 0, one finds immediately that the number of molecules whose absolute value of velocities lies between c and c + dc is given approximately by Eq. (12). Equation (12)
is useful in many applications.
If, on the other hand, a gas is in a uniform gravitational field so that U = −mgz, one finds again from Eqs. (9a) and (9b) that the number of molecules at the height z (irrespective of their velocities) is given by notation (13).
If one uses the fact that β = 1/kT, then notation (13) expresses the famous barometric density formula, the density distribution of an ideal gas in a constant gravitational field.
Statistical method; fluctuations
The indiscriminate use of the collision-number assumption leads, via the H theorem, to paradoxical results. The basic conflict stems from the irreversible results that appear to emerge as a consequence of a large number of reversible fundamental processes. Although a complete detailed reconciliation between the irreversible aspects of thermodynamics as a consequence of reversible dynamics remains to be given, it is now well understood that a careful treatment of the explicit and hidden probability assumptions is the key to the understanding of the apparent conflict. It is clear, for instance, that Eq. (2) is to be understood either in an average sense (the average number of collisions is given by klij, where ni nj is an average of the product) or as a probability, but not as a definite causal law. The treatment usually given actually presupposes that ninj = ; that is, it neglects fluctuations. See also: Chemical thermodynamics
To introduce probability ideas in a more explicit manner, consider a distribution of N molecules over the cells wi in μ space, as shown in notation (14).
Let the total volume of the μ space be Ω. Suppose that N points are thrown into the μ space. What is the probability that just n1 points will end up in cell 1, n2 in cell 2, ni in cell i, and so on? If all points in the space are equally likely (equal a priori probabilities), the probability of such an occurrence is shown in Eq. (15).
One now really defines the probability of a physical state characterized by notation (14) as given by the essentially geometric probability shown in Eq. (15). The probability of a state so defined is intimately connected with the H function. If one takes the ln of Eq. (15) and applies Stirling's approximation for ln n!, one obtains Eq. (16).
The H theorem, which states that H decreases in the course of time, therefore may be rephrased to state that in the course of time the system evolves toward a more probable state. Actually the H function may be related (for equilibrium states only) to the thermodynamic entropy S by Eq. (17).
Here k is, again, the Boltzmann constant.
which relates the entropy of a state to the probability of that state, sometimes called Boltzmann's relation. See also: Probability
The state of maximum probability, that is, the set of occupation numbers n1, … , ni … , which maximizes Eq. (15), subject to the auxiliary conditions given in Eqs. (10a) and (10b), turns out to be given by the special distribution written as Eqs. (19a), (19b), and (19c).
This again is the Maxwell-Boltzmann distribution. Hence the equilibrium distribution may be thought of as the most probable state of a system. If a system is not in equilibrium, it will most likely (but not certainly) go there; if it is in equilibrium, it will most likely (but not certainly) stay there. By using such probability statements, it may be shown that the paradoxes and conflicts mentioned before may indeed be removed. The general situation is still not clear in all details, although much progress has been made, especially for simple models. A consequence of the probabilistic character of statistics is that the entities computed also possess this characteristic. For example, one cannot really speak definitively of the number of molecules hitting a section of the wall per second, but only about the probability that a given number will hit the wall, or about the average number hitting. In the same vein, the amount of momentum transferred to a unit area of the wall by the molecules per second (this, in fact, is precisely the pressure) is also to be understood as an average. This in particular means that the pressure is a fluctuating entity. In general, it may be shown that the fluctuations in a quantity Q are defined by expression (20),
where = the average of Q, and N = number of particles in the systems. When the fluctuations are small, the statistical description as well as the thermodynamic concept is useful. The fluctuations in pressure may be demonstrated by observing the motion of a mirror, suspended by a fiber, in a gas. On the average, as many gas molecules will hit the back as the front of the mirror, so that the average displacement will indeed be zero. However, it is easy to imagine a situation where more momentum is transferred in one direction than in another, resulting in a deflection of the mirror. From the knowledge of the distribution function the probabilities for such occurrences may indeed be computed; the calculated and observed behavior agree very well. This clearly demonstrates the essentially statistical character of the pressure. See also: Boltzmann transport equation; Brownian movement; Kinetic theory of matter; Quantum statistics; Statistical mechanics